Question

A particle of mass 10g is kept on the surface of a uniform sphere of mass, 100kg and radius 10 cm be done against the gravitational force between them to take the particle far away from the sphere $(G=6.67\times {{10}^{-11}}N{{m}^{2}}k{{g}^{-2}})$

Answer 1

Firstly we apply formula for Gravitational potential energy of the two bodies when initially both are placed on each other and then finally we apply the relation when they are far apart .By default we consider the far part means upto infinity. Then we apply the work energy theorem so that we can get the required value of work done.

Complete step-by-step answer:
Since in the question it is given that a particle of mass 10g is kept on the surface of a uniform sphere of mass 100kg whose radius is 10cm. Then gravitational potential energy is acting between them which hold both the particles of the system together. Here we have to calculate the work done against the gravitational force when we take particles far away from the sphere upto infinity. By default we consider far away means upto infinity.
Initially particle of mass 10g is on the surface of uniform sphere of mass 100kg so initial gravitational potential energy of the system is represented as${{U}_{i}}$: -
${{U}_{i}}=-\dfrac{GMm}{r}$(Equation 1)
Here mass of particle is represented by m=10g$$$$=10\times {{10}^{-3}}Kg
Mass of the sphere is represented by$M=100Kg$.
Distance between mass of particle and centre of sphere is represented by $r=10cm=.1m$
Now, put these value in the equation 1 and we get,
$\Rightarrow {{U}_{i}}=-\dfrac{6.67\times {{10}^{-11}}\times 100\times .01}{0.1}$
$\therefore {{U}_{i}}=-6.67\times {{10}^{-10}}J$
Negative sign shows that one body is bound with another body with some gravitational force of attraction.

Now this particle of mass 10g is taken far away from the centre of the sphere, so it is taken to infinity.
So at infinity, Gravitational Potential Energy becomes 0.
$\therefore {{U}_{f}}=0$
According to the work energy theorem, we know that change in Gravitational potential energy is equal to work done by a particle in moving from surface to infinity against gravitational force.
Work can only be done when it is against some force. So work done can be written as,
$W={{U}_{f}}-{{U}_{i}}$
Put the value of Initial and final gravitational potential energy, we get

& \Rightarrow W=0-(-6.67\times {{10}^{-10}}) \\\ & \therefore W=+6.67\times {{10}^{-10}}J \\\ \end{aligned}$$ Positive work done means that this amount of energy is required to do the work on a particle so that it can reach to infinity. **Note:** Work can only be exist in nature when we are displacing anybody against some resistive force because in that case we have to apply some effort to move them so work is required .Like when body falls from height towards earth then no work is done due to gravitational forces but that body is facing some resistive force due to air so work is only be done for that resistive forces.